LMFDB - Embedded newform 8048.2.a.w.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$29$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.22567 q^{3} -3.22615 q^{5} -0.00440441 q^{7} +1.95359 q^{9} +O(q^{10})$	$q-2.22567 q^{3} -3.22615 q^{5} -0.00440441 q^{7} +1.95359 q^{9} -1.22033 q^{11} +4.82592 q^{13} +7.18034 q^{15} -2.43808 q^{17} +7.24602 q^{19} +0.00980273 q^{21} +4.27079 q^{23} +5.40806 q^{25} +2.32897 q^{27} -2.39478 q^{29} -5.78267 q^{31} +2.71605 q^{33} +0.0142093 q^{35} +2.84908 q^{37} -10.7409 q^{39} +0.665610 q^{41} -2.23735 q^{43} -6.30257 q^{45} +6.13063 q^{47} -6.99998 q^{49} +5.42635 q^{51} -13.9656 q^{53} +3.93698 q^{55} -16.1272 q^{57} +8.00396 q^{59} +15.0576 q^{61} -0.00860439 q^{63} -15.5692 q^{65} +8.09692 q^{67} -9.50535 q^{69} +2.79160 q^{71} +1.66417 q^{73} -12.0365 q^{75} +0.00537485 q^{77} -3.84118 q^{79} -11.0443 q^{81} -8.07946 q^{83} +7.86561 q^{85} +5.32998 q^{87} -5.05194 q^{89} -0.0212553 q^{91} +12.8703 q^{93} -23.3768 q^{95} -4.28267 q^{97} -2.38403 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.22567	−1.28499	−0.642494	−	0.766291i	$-0.722100\pi$
$3$	−2.22567	−1.28499	−0.642494	+	0.766291i	$0.722100\pi$
$4$	0	0
$5$	−3.22615	−1.44278	−0.721390	−	0.692529i	$-0.756496\pi$
$5$	−3.22615	−1.44278	−0.721390	+	0.692529i	$0.756496\pi$
$6$	0	0
$7$	−0.00440441	−0.00166471	−0.000832355	−	1.00000i	$-0.500265\pi$
$7$	−0.00440441	−0.00166471	−0.000832355		1.00000i	$0.500265\pi$
$8$	0	0
$9$	1.95359	0.651195
$10$	0	0
$11$	−1.22033	−0.367944	−0.183972	−	0.982931i	$-0.558896\pi$
$11$	−1.22033	−0.367944	−0.183972	+	0.982931i	$0.558896\pi$
$12$	0	0
$13$	4.82592	1.33847	0.669234	−	0.743051i	$-0.266622\pi$
$13$	4.82592	1.33847	0.669234	+	0.743051i	$0.266622\pi$
$14$	0	0
$15$	7.18034	1.85396
$16$	0	0
$17$	−2.43808	−0.591321	−0.295660	−	0.955293i	$-0.595540\pi$
$17$	−2.43808	−0.591321	−0.295660	+	0.955293i	$0.595540\pi$
$18$	0	0
$19$	7.24602	1.66235	0.831176	−	0.556010i	$-0.187668\pi$
$19$	7.24602	1.66235	0.831176	+	0.556010i	$0.187668\pi$
$20$	0	0
$21$	0.00980273	0.00213913
$22$	0	0
$23$	4.27079	0.890522	0.445261	−	0.895401i	$-0.353111\pi$
$23$	4.27079	0.890522	0.445261	+	0.895401i	$0.353111\pi$
$24$	0	0
$25$	5.40806	1.08161
$26$	0	0
$27$	2.32897	0.448210
$28$	0	0
$29$	−2.39478	−0.444699	−0.222350	−	0.974967i	$-0.571373\pi$
$29$	−2.39478	−0.444699	−0.222350	+	0.974967i	$0.571373\pi$
$30$	0	0
$31$	−5.78267	−1.03860	−0.519299	−	0.854593i	$-0.673807\pi$
$31$	−5.78267	−1.03860	−0.519299	+	0.854593i	$0.673807\pi$
$32$	0	0
$33$	2.71605	0.472804
$34$	0	0
$35$	0.0142093	0.00240181
$36$	0	0
$37$	2.84908	0.468386	0.234193	−	0.972190i	$-0.424755\pi$
$37$	2.84908	0.468386	0.234193	+	0.972190i	$0.424755\pi$
$38$	0	0
$39$	−10.7409	−1.71992
$40$	0	0
$41$	0.665610	0.103951	0.0519754	−	0.998648i	$-0.483448\pi$
$41$	0.665610	0.103951	0.0519754	+	0.998648i	$0.483448\pi$
$42$	0	0
$43$	−2.23735	−0.341193	−0.170597	−	0.985341i	$-0.554570\pi$
$43$	−2.23735	−0.341193	−0.170597	+	0.985341i	$0.554570\pi$
$44$	0	0
$45$	−6.30257	−0.939531
$46$	0	0
$47$	6.13063	0.894245	0.447122	−	0.894473i	$-0.352449\pi$
$47$	6.13063	0.894245	0.447122	+	0.894473i	$0.352449\pi$
$48$	0	0
$49$	−6.99998	−0.999997
$50$	0	0
$51$	5.42635	0.759841
$52$	0	0
$53$	−13.9656	−1.91832	−0.959162	−	0.282859i	$-0.908717\pi$
$53$	−13.9656	−1.91832	−0.959162	+	0.282859i	$0.908717\pi$
$54$	0	0
$55$	3.93698	0.530863
$56$	0	0
$57$	−16.1272	−2.13610
$58$	0	0
$59$	8.00396	1.04203	0.521013	−	0.853549i	$-0.325554\pi$
$59$	8.00396	1.04203	0.521013	+	0.853549i	$0.325554\pi$
$60$	0	0
$61$	15.0576	1.92793	0.963967	−	0.266021i	$-0.0857092\pi$
$61$	15.0576	1.92793	0.963967	+	0.266021i	$0.0857092\pi$
$62$	0	0
$63$	−0.00860439	−0.00108405
$64$	0	0
$65$	−15.5692	−1.93112
$66$	0	0
$67$	8.09692	0.989197	0.494598	−	0.869122i	$-0.335315\pi$
$67$	8.09692	0.989197	0.494598	+	0.869122i	$0.335315\pi$
$68$	0	0
$69$	−9.50535	−1.14431
$70$	0	0
$71$	2.79160	0.331302	0.165651	−	0.986184i	$-0.447027\pi$
$71$	2.79160	0.331302	0.165651	+	0.986184i	$0.447027\pi$
$72$	0	0
$73$	1.66417	0.194776	0.0973881	−	0.995246i	$-0.468951\pi$
$73$	1.66417	0.194776	0.0973881	+	0.995246i	$0.468951\pi$
$74$	0	0
$75$	−12.0365	−1.38986
$76$	0	0
$77$	0.00537485	0.000612520 0
$78$	0	0
$79$	−3.84118	−0.432167	−0.216083	−	0.976375i	$-0.569328\pi$
$79$	−3.84118	−0.432167	−0.216083	+	0.976375i	$0.569328\pi$
$80$	0	0
$81$	−11.0443	−1.22714
$82$	0	0
$83$	−8.07946	−0.886836	−0.443418	−	0.896315i	$-0.646234\pi$
$83$	−8.07946	−0.886836	−0.443418	+	0.896315i	$0.646234\pi$
$84$	0	0
$85$	7.86561	0.853146
$86$	0	0
$87$	5.32998	0.571433
$88$	0	0
$89$	−5.05194	−0.535505	−0.267752	−	0.963488i	$-0.586281\pi$
$89$	−5.05194	−0.535505	−0.267752	+	0.963488i	$0.586281\pi$
$90$	0	0
$91$	−0.0212553	−0.00222816
$92$	0	0
$93$	12.8703	1.33459
$94$	0	0
$95$	−23.3768	−2.39841
$96$	0	0
$97$	−4.28267	−0.434839	−0.217420	−	0.976078i	$-0.569764\pi$
$97$	−4.28267	−0.434839	−0.217420	+	0.976078i	$0.569764\pi$
$98$	0	0
$99$	−2.38403	−0.239604
$100$	0	0
$101$	−9.63378	−0.958597	−0.479299	−	0.877652i	$-0.659109\pi$
$101$	−9.63378	−0.958597	−0.479299	+	0.877652i	$0.659109\pi$
$102$	0	0
$103$	3.01370	0.296949	0.148474	−	0.988916i	$-0.452564\pi$
$103$	3.01370	0.296949	0.148474	+	0.988916i	$0.452564\pi$
$104$	0	0
$105$	−0.0316251	−0.00308630
$106$	0	0
$107$	−18.4523	−1.78385	−0.891924	−	0.452186i	$-0.850644\pi$
$107$	−18.4523	−1.78385	−0.891924	+	0.452186i	$0.850644\pi$
$108$	0	0
$109$	−12.6100	−1.20782	−0.603912	−	0.797051i	$-0.706392\pi$
$109$	−12.6100	−1.20782	−0.603912	+	0.797051i	$0.706392\pi$
$110$	0	0
$111$	−6.34110	−0.601870
$112$	0	0
$113$	−3.13577	−0.294988	−0.147494	−	0.989063i	$-0.547121\pi$
$113$	−3.13577	−0.294988	−0.147494	+	0.989063i	$0.547121\pi$
$114$	0	0
$115$	−13.7782	−1.28483
$116$	0	0
$117$	9.42785	0.871605
$118$	0	0
$119$	0.0107383	0.000984377 0
$120$	0	0
$121$	−9.51079	−0.864617
$122$	0	0
$123$	−1.48142	−0.133576
$124$	0	0
$125$	−1.31647	−0.117749
$126$	0	0
$127$	−7.75522	−0.688165	−0.344082	−	0.938939i	$-0.611810\pi$
$127$	−7.75522	−0.688165	−0.344082	+	0.938939i	$0.611810\pi$
$128$	0	0
$129$	4.97960	0.438429
$130$	0	0
$131$	15.2272	1.33040	0.665202	−	0.746664i	$-0.268346\pi$
$131$	15.2272	1.33040	0.665202	+	0.746664i	$0.268346\pi$
$132$	0	0
$133$	−0.0319144	−0.00276733
$134$	0	0
$135$	−7.51360	−0.646668
$136$	0	0
$137$	−18.3171	−1.56494	−0.782469	−	0.622690i	$-0.786040\pi$
$137$	−18.3171	−1.56494	−0.782469	+	0.622690i	$0.786040\pi$
$138$	0	0
$139$	4.15381	0.352321	0.176161	−	0.984361i	$-0.443632\pi$
$139$	4.15381	0.352321	0.176161	+	0.984361i	$0.443632\pi$
$140$	0	0
$141$	−13.6447	−1.14909
$142$	0	0
$143$	−5.88923	−0.492482
$144$	0	0
$145$	7.72592	0.641603
$146$	0	0
$147$	15.5796	1.28498
$148$	0	0
$149$	13.5549	1.11046	0.555231	−	0.831696i	$-0.312630\pi$
$149$	13.5549	1.11046	0.555231	+	0.831696i	$0.312630\pi$
$150$	0	0
$151$	10.0500	0.817859	0.408929	−	0.912566i	$-0.365902\pi$
$151$	10.0500	0.817859	0.408929	+	0.912566i	$0.365902\pi$
$152$	0	0
$153$	−4.76300	−0.385065
$154$	0	0
$155$	18.6558	1.49847
$156$	0	0
$157$	−16.2058	−1.29337	−0.646684	−	0.762758i	$-0.723845\pi$
$157$	−16.2058	−1.29337	−0.646684	+	0.762758i	$0.723845\pi$
$158$	0	0
$159$	31.0828	2.46502
$160$	0	0
$161$	−0.0188103	−0.00148246
$162$	0	0
$163$	23.3541	1.82923	0.914615	−	0.404325i	$-0.132494\pi$
$163$	23.3541	1.82923	0.914615	+	0.404325i	$0.132494\pi$
$164$	0	0
$165$	−8.76241	−0.682152
$166$	0	0
$167$	−0.513742	−0.0397546	−0.0198773	−	0.999802i	$-0.506328\pi$
$167$	−0.513742	−0.0397546	−0.0198773	+	0.999802i	$0.506328\pi$
$168$	0	0
$169$	10.2895	0.791499
$170$	0	0
$171$	14.1557	1.08252
$172$	0	0
$173$	0.0555917	0.00422656	0.00211328	−	0.999998i	$-0.499327\pi$
$173$	0.0555917	0.00422656	0.00211328	+	0.999998i	$0.499327\pi$
$174$	0	0
$175$	−0.0238193	−0.00180057
$176$	0	0
$177$	−17.8141	−1.33899
$178$	0	0
$179$	24.9936	1.86811	0.934055	−	0.357129i	$-0.116244\pi$
$179$	24.9936	1.86811	0.934055	+	0.357129i	$0.116244\pi$
$180$	0	0
$181$	2.61076	0.194057	0.0970283	−	0.995282i	$-0.469066\pi$
$181$	2.61076	0.194057	0.0970283	+	0.995282i	$0.469066\pi$
$182$	0	0
$183$	−33.5133	−2.47737
$184$	0	0
$185$	−9.19156	−0.675777
$186$	0	0
$187$	2.97527	0.217573
$188$	0	0
$189$	−0.0102577	−0.000746139 0
$190$	0	0
$191$	−25.0221	−1.81053	−0.905267	−	0.424843i	$-0.860329\pi$
$191$	−25.0221	−1.81053	−0.905267	+	0.424843i	$0.860329\pi$
$192$	0	0
$193$	11.0412	0.794764	0.397382	−	0.917653i	$-0.369919\pi$
$193$	11.0412	0.794764	0.397382	+	0.917653i	$0.369919\pi$
$194$	0	0
$195$	34.6517	2.48146
$196$	0	0
$197$	19.9589	1.42201	0.711006	−	0.703186i	$-0.248240\pi$
$197$	19.9589	1.42201	0.711006	+	0.703186i	$0.248240\pi$
$198$	0	0
$199$	3.44841	0.244451	0.122226	−	0.992502i	$-0.460997\pi$
$199$	3.44841	0.244451	0.122226	+	0.992502i	$0.460997\pi$
$200$	0	0
$201$	−18.0210	−1.27111
$202$	0	0
$203$	0.0105476	0.000740295 0
$204$	0	0
$205$	−2.14736	−0.149978
$206$	0	0
$207$	8.34336	0.579904
$208$	0	0
$209$	−8.84256	−0.611653
$210$	0	0
$211$	1.43339	0.0986785	0.0493393	−	0.998782i	$-0.484288\pi$
$211$	1.43339	0.0986785	0.0493393	+	0.998782i	$0.484288\pi$
$212$	0	0
$213$	−6.21318	−0.425720
$214$	0	0
$215$	7.21805	0.492267
$216$	0	0
$217$	0.0254692	0.00172896
$218$	0	0
$219$	−3.70388	−0.250285
$220$	0	0
$221$	−11.7660	−0.791465
$222$	0	0
$223$	0.982215	0.0657740	0.0328870	−	0.999459i	$-0.489530\pi$
$223$	0.982215	0.0657740	0.0328870	+	0.999459i	$0.489530\pi$
$224$	0	0
$225$	10.5651	0.704341
$226$	0	0
$227$	17.4830	1.16039	0.580194	−	0.814478i	$-0.302977\pi$
$227$	17.4830	1.16039	0.580194	+	0.814478i	$0.302977\pi$
$228$	0	0
$229$	−5.92218	−0.391349	−0.195674	−	0.980669i	$-0.562690\pi$
$229$	−5.92218	−0.391349	−0.195674	+	0.980669i	$0.562690\pi$
$230$	0	0
$231$	−0.0119626	−0.000787082 0
$232$	0	0
$233$	29.1710	1.91106	0.955528	−	0.294900i	$-0.0952864\pi$
$233$	29.1710	1.91106	0.955528	+	0.294900i	$0.0952864\pi$
$234$	0	0
$235$	−19.7784	−1.29020
$236$	0	0
$237$	8.54919	0.555329
$238$	0	0
$239$	15.6900	1.01490	0.507450	−	0.861681i	$-0.330588\pi$
$239$	15.6900	1.01490	0.507450	+	0.861681i	$0.330588\pi$
$240$	0	0
$241$	5.16704	0.332838	0.166419	−	0.986055i	$-0.446780\pi$
$241$	5.16704	0.332838	0.166419	+	0.986055i	$0.446780\pi$
$242$	0	0
$243$	17.5939	1.12865
$244$	0	0
$245$	22.5830	1.44278
$246$	0	0
$247$	34.9687	2.22501
$248$	0	0
$249$	17.9822	1.13957
$250$	0	0
$251$	−20.5126	−1.29474	−0.647372	−	0.762174i	$-0.724132\pi$
$251$	−20.5126	−1.29474	−0.647372	+	0.762174i	$0.724132\pi$
$252$	0	0
$253$	−5.21179	−0.327662
$254$	0	0
$255$	−17.5062	−1.09628
$256$	0	0
$257$	−27.9525	−1.74363	−0.871815	−	0.489835i	$-0.837057\pi$
$257$	−27.9525	−1.74363	−0.871815	+	0.489835i	$0.837057\pi$
$258$	0	0
$259$	−0.0125485	−0.000779726 0
$260$	0	0
$261$	−4.67841	−0.289586
$262$	0	0
$263$	11.2349	0.692775	0.346388	−	0.938092i	$-0.387408\pi$
$263$	11.2349	0.692775	0.346388	+	0.938092i	$0.387408\pi$
$264$	0	0
$265$	45.0552	2.76772
$266$	0	0
$267$	11.2439	0.688118
$268$	0	0
$269$	−15.8912	−0.968901	−0.484450	−	0.874819i	$-0.660980\pi$
$269$	−15.8912	−0.968901	−0.484450	+	0.874819i	$0.660980\pi$
$270$	0	0
$271$	19.2859	1.17153	0.585766	−	0.810480i	$-0.300794\pi$
$271$	19.2859	1.17153	0.585766	+	0.810480i	$0.300794\pi$
$272$	0	0
$273$	0.0473072	0.00286316
$274$	0	0
$275$	−6.59964	−0.397973
$276$	0	0
$277$	18.8119	1.13029	0.565147	−	0.824990i	$-0.308819\pi$
$277$	18.8119	1.13029	0.565147	+	0.824990i	$0.308819\pi$
$278$	0	0
$279$	−11.2969	−0.676330
$280$	0	0
$281$	6.90859	0.412132	0.206066	−	0.978538i	$-0.433934\pi$
$281$	6.90859	0.412132	0.206066	+	0.978538i	$0.433934\pi$
$282$	0	0
$283$	−16.1784	−0.961705	−0.480853	−	0.876801i	$-0.659673\pi$
$283$	−16.1784	−0.961705	−0.480853	+	0.876801i	$0.659673\pi$
$284$	0	0
$285$	52.0289	3.08192
$286$	0	0
$287$	−0.00293162	−0.000173048 0
$288$	0	0
$289$	−11.0558	−0.650340
$290$	0	0
$291$	9.53179	0.558763
$292$	0	0
$293$	−15.9351	−0.930936	−0.465468	−	0.885065i	$-0.654114\pi$
$293$	−15.9351	−0.930936	−0.465468	+	0.885065i	$0.654114\pi$
$294$	0	0
$295$	−25.8220	−1.50341
$296$	0	0
$297$	−2.84212	−0.164916
$298$	0	0
$299$	20.6105	1.19194
$300$	0	0
$301$	0.00985422	0.000567988 0
$302$	0	0
$303$	21.4416	1.23179
$304$	0	0
$305$	−48.5783	−2.78158
$306$	0	0
$307$	−4.97279	−0.283812	−0.141906	−	0.989880i	$-0.545323\pi$
$307$	−4.97279	−0.283812	−0.141906	+	0.989880i	$0.545323\pi$
$308$	0	0
$309$	−6.70749	−0.381576
$310$	0	0
$311$	−12.4927	−0.708398	−0.354199	−	0.935170i	$-0.615246\pi$
$311$	−12.4927	−0.708398	−0.354199	+	0.935170i	$0.615246\pi$
$312$	0	0
$313$	−17.6757	−0.999089	−0.499545	−	0.866288i	$-0.666499\pi$
$313$	−17.6757	−0.999089	−0.499545	+	0.866288i	$0.666499\pi$
$314$	0	0
$315$	0.0277591	0.00156405
$316$	0	0
$317$	26.8469	1.50787	0.753935	−	0.656949i	$-0.228153\pi$
$317$	26.8469	1.50787	0.753935	+	0.656949i	$0.228153\pi$
$318$	0	0
$319$	2.92243	0.163625
$320$	0	0
$321$	41.0686	2.29222
$322$	0	0
$323$	−17.6664	−0.982983
$324$	0	0
$325$	26.0989	1.44770
$326$	0	0
$327$	28.0657	1.55204
$328$	0	0
$329$	−0.0270018	−0.00148866
$330$	0	0
$331$	−7.45932	−0.410001	−0.205001	−	0.978762i	$-0.565720\pi$
$331$	−7.45932	−0.410001	−0.205001	+	0.978762i	$0.565720\pi$
$332$	0	0
$333$	5.56592	0.305011
$334$	0	0
$335$	−26.1219	−1.42719
$336$	0	0
$337$	−2.26145	−0.123189	−0.0615945	−	0.998101i	$-0.519619\pi$
$337$	−2.26145	−0.123189	−0.0615945	+	0.998101i	$0.519619\pi$
$338$	0	0
$339$	6.97917	0.379057
$340$	0	0
$341$	7.05678	0.382146
$342$	0	0
$343$	0.0616616	0.00332941
$344$	0	0
$345$	30.6657	1.65099
$346$	0	0
$347$	3.67419	0.197241	0.0986205	−	0.995125i	$-0.468557\pi$
$347$	3.67419	0.197241	0.0986205	+	0.995125i	$0.468557\pi$
$348$	0	0
$349$	19.8348	1.06173	0.530867	−	0.847455i	$-0.321866\pi$
$349$	19.8348	1.06173	0.530867	+	0.847455i	$0.321866\pi$
$350$	0	0
$351$	11.2394	0.599915
$352$	0	0
$353$	−5.27900	−0.280973	−0.140486	−	0.990083i	$-0.544867\pi$
$353$	−5.27900	−0.280973	−0.140486	+	0.990083i	$0.544867\pi$
$354$	0	0
$355$	−9.00614	−0.477996
$356$	0	0
$357$	−0.0238998	−0.00126491
$358$	0	0
$359$	−23.3733	−1.23359	−0.616797	−	0.787122i	$-0.711570\pi$
$359$	−23.3733	−1.23359	−0.616797	+	0.787122i	$0.711570\pi$
$360$	0	0
$361$	33.5048	1.76341
$362$	0	0
$363$	21.1678	1.11102
$364$	0	0
$365$	−5.36886	−0.281019
$366$	0	0
$367$	−26.0030	−1.35734	−0.678672	−	0.734442i	$-0.737444\pi$
$367$	−26.0030	−1.35734	−0.678672	+	0.734442i	$0.737444\pi$
$368$	0	0
$369$	1.30033	0.0676923
$370$	0	0
$371$	0.0615102	0.00319345
$372$	0	0
$373$	7.34064	0.380084	0.190042	−	0.981776i	$-0.439138\pi$
$373$	7.34064	0.380084	0.190042	+	0.981776i	$0.439138\pi$
$374$	0	0
$375$	2.93003	0.151306
$376$	0	0
$377$	−11.5570	−0.595216
$378$	0	0
$379$	7.31495	0.375743	0.187872	−	0.982194i	$-0.439841\pi$
$379$	7.31495	0.375743	0.187872	+	0.982194i	$0.439841\pi$
$380$	0	0
$381$	17.2605	0.884284
$382$	0	0
$383$	1.15583	0.0590603	0.0295302	−	0.999564i	$-0.490599\pi$
$383$	1.15583	0.0590603	0.0295302	+	0.999564i	$0.490599\pi$
$384$	0	0
$385$	−0.0173401	−0.000883732 0
$386$	0	0
$387$	−4.37086	−0.222183
$388$	0	0
$389$	31.7388	1.60922	0.804609	−	0.593804i	$-0.202375\pi$
$389$	31.7388	1.60922	0.804609	+	0.593804i	$0.202375\pi$
$390$	0	0
$391$	−10.4125	−0.526584
$392$	0	0
$393$	−33.8906	−1.70955
$394$	0	0
$395$	12.3922	0.623521
$396$	0	0
$397$	18.4702	0.926995	0.463497	−	0.886098i	$-0.346594\pi$
$397$	18.4702	0.926995	0.463497	+	0.886098i	$0.346594\pi$
$398$	0	0
$399$	0.0710308	0.00355599
$400$	0	0
$401$	−17.3782	−0.867824	−0.433912	−	0.900955i	$-0.642867\pi$
$401$	−17.3782	−0.867824	−0.433912	+	0.900955i	$0.642867\pi$
$402$	0	0
$403$	−27.9067	−1.39013
$404$	0	0
$405$	35.6305	1.77049
$406$	0	0
$407$	−3.47683	−0.172340
$408$	0	0
$409$	22.2224	1.09883	0.549413	−	0.835551i	$-0.314852\pi$
$409$	22.2224	1.09883	0.549413	+	0.835551i	$0.314852\pi$
$410$	0	0
$411$	40.7678	2.01093
$412$	0	0
$413$	−0.0352527	−0.00173467
$414$	0	0
$415$	26.0656	1.27951
$416$	0	0
$417$	−9.24498	−0.452729
$418$	0	0
$419$	−15.6559	−0.764841	−0.382420	−	0.923988i	$-0.624909\pi$
$419$	−15.6559	−0.764841	−0.382420	+	0.923988i	$0.624909\pi$
$420$	0	0
$421$	38.6873	1.88550	0.942752	−	0.333493i	$-0.108227\pi$
$421$	38.6873	1.88550	0.942752	+	0.333493i	$0.108227\pi$
$422$	0	0
$423$	11.9767	0.582328
$424$	0	0
$425$	−13.1853	−0.639580
$426$	0	0
$427$	−0.0663200	−0.00320945
$428$	0	0
$429$	13.1075	0.632834
$430$	0	0
$431$	−16.8154	−0.809971	−0.404985	−	0.914323i	$-0.632723\pi$
$431$	−16.8154	−0.809971	−0.404985	+	0.914323i	$0.632723\pi$
$432$	0	0
$433$	32.0543	1.54043	0.770217	−	0.637782i	$-0.220148\pi$
$433$	32.0543	1.54043	0.770217	+	0.637782i	$0.220148\pi$
$434$	0	0
$435$	−17.1953	−0.824452
$436$	0	0
$437$	30.9462	1.48036
$438$	0	0
$439$	−10.3720	−0.495030	−0.247515	−	0.968884i	$-0.579614\pi$
$439$	−10.3720	−0.495030	−0.247515	+	0.968884i	$0.579614\pi$
$440$	0	0
$441$	−13.6751	−0.651194
$442$	0	0
$443$	16.6682	0.791930	0.395965	−	0.918266i	$-0.370410\pi$
$443$	16.6682	0.791930	0.395965	+	0.918266i	$0.370410\pi$
$444$	0	0
$445$	16.2983	0.772615
$446$	0	0
$447$	−30.1687	−1.42693
$448$	0	0
$449$	−2.56585	−0.121090	−0.0605449	−	0.998165i	$-0.519284\pi$
$449$	−2.56585	−0.121090	−0.0605449	+	0.998165i	$0.519284\pi$
$450$	0	0
$451$	−0.812266	−0.0382481
$452$	0	0
$453$	−22.3680	−1.05094
$454$	0	0
$455$	0.0685729	0.00321475
$456$	0	0
$457$	−4.29783	−0.201044	−0.100522	−	0.994935i	$-0.532051\pi$
$457$	−4.29783	−0.201044	−0.100522	+	0.994935i	$0.532051\pi$
$458$	0	0
$459$	−5.67820	−0.265036
$460$	0	0
$461$	−5.30365	−0.247016	−0.123508	−	0.992344i	$-0.539414\pi$
$461$	−5.30365	−0.247016	−0.123508	+	0.992344i	$0.539414\pi$
$462$	0	0
$463$	33.4640	1.55521	0.777603	−	0.628755i	$-0.216435\pi$
$463$	33.4640	1.55521	0.777603	+	0.628755i	$0.216435\pi$
$464$	0	0
$465$	−41.5215	−1.92551
$466$	0	0
$467$	14.1463	0.654612	0.327306	−	0.944918i	$-0.393859\pi$
$467$	14.1463	0.654612	0.327306	+	0.944918i	$0.393859\pi$
$468$	0	0
$469$	−0.0356621	−0.00164672
$470$	0	0
$471$	36.0688	1.66196
$472$	0	0
$473$	2.73032	0.125540
$474$	0	0
$475$	39.1869	1.79802
$476$	0	0
$477$	−27.2830	−1.24920
$478$	0	0
$479$	−2.29912	−0.105050	−0.0525248	−	0.998620i	$-0.516727\pi$
$479$	−2.29912	−0.105050	−0.0525248	+	0.998620i	$0.516727\pi$
$480$	0	0
$481$	13.7494	0.626920
$482$	0	0
$483$	0.0418654	0.00190494
$484$	0	0
$485$	13.8165	0.627377
$486$	0	0
$487$	18.7332	0.848881	0.424441	−	0.905456i	$-0.360471\pi$
$487$	18.7332	0.848881	0.424441	+	0.905456i	$0.360471\pi$
$488$	0	0
$489$	−51.9783	−2.35054
$490$	0	0
$491$	−16.7820	−0.757361	−0.378681	−	0.925527i	$-0.623622\pi$
$491$	−16.7820	−0.757361	−0.378681	+	0.925527i	$0.623622\pi$
$492$	0	0
$493$	5.83866	0.262960
$494$	0	0
$495$	7.69124	0.345695
$496$	0	0
$497$	−0.0122954	−0.000551522 0
$498$	0	0
$499$	22.7726	1.01944	0.509722	−	0.860339i	$-0.329748\pi$
$499$	22.7726	1.01944	0.509722	+	0.860339i	$0.329748\pi$
$500$	0	0
$501$	1.14342	0.0510842
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	31.0801	1.38304
$506$	0	0
$507$	−22.9010	−1.01707
$508$	0	0
$509$	5.02500	0.222729	0.111365	−	0.993780i	$-0.464478\pi$
$509$	5.02500	0.222729	0.111365	+	0.993780i	$0.464478\pi$
$510$	0	0
$511$	−0.00732967	−0.000324246 0
$512$	0	0
$513$	16.8757	0.745082
$514$	0	0
$515$	−9.72266	−0.428432
$516$	0	0
$517$	−7.48142	−0.329032
$518$	0	0
$519$	−0.123729	−0.00543108
$520$	0	0
$521$	4.93410	0.216167	0.108084	−	0.994142i	$-0.465529\pi$
$521$	4.93410	0.216167	0.108084	+	0.994142i	$0.465529\pi$
$522$	0	0
$523$	−31.3795	−1.37213	−0.686066	−	0.727539i	$-0.740664\pi$
$523$	−31.3795	−1.37213	−0.686066	+	0.727539i	$0.740664\pi$
$524$	0	0
$525$	0.0530138	0.00231371
$526$	0	0
$527$	14.0986	0.614144
$528$	0	0
$529$	−4.76034	−0.206971
$530$	0	0
$531$	15.6364	0.678563
$532$	0	0
$533$	3.21218	0.139135
$534$	0	0
$535$	59.5298	2.57370
$536$	0	0
$537$	−55.6274	−2.40050
$538$	0	0
$539$	8.54231	0.367943
$540$	0	0
$541$	7.84946	0.337475	0.168737	−	0.985661i	$-0.446031\pi$
$541$	7.84946	0.337475	0.168737	+	0.985661i	$0.446031\pi$
$542$	0	0
$543$	−5.81069	−0.249360
$544$	0	0
$545$	40.6819	1.74262
$546$	0	0
$547$	11.1689	0.477548	0.238774	−	0.971075i	$-0.423255\pi$
$547$	11.1689	0.477548	0.238774	+	0.971075i	$0.423255\pi$
$548$	0	0
$549$	29.4164	1.25546
$550$	0	0
$551$	−17.3526	−0.739246
$552$	0	0
$553$	0.0169181	0.000719432 0
$554$	0	0
$555$	20.4573	0.868366
$556$	0	0
$557$	−44.6007	−1.88979	−0.944896	−	0.327371i	$-0.893837\pi$
$557$	−44.6007	−1.88979	−0.944896	+	0.327371i	$0.893837\pi$
$558$	0	0
$559$	−10.7973	−0.456677
$560$	0	0
$561$	−6.62195	−0.279579
$562$	0	0
$563$	22.7801	0.960067	0.480033	−	0.877250i	$-0.340625\pi$
$563$	22.7801	0.960067	0.480033	+	0.877250i	$0.340625\pi$
$564$	0	0
$565$	10.1165	0.425603
$566$	0	0
$567$	0.0486434	0.00204283
$568$	0	0
$569$	−27.0595	−1.13439	−0.567197	−	0.823582i	$-0.691972\pi$
$569$	−27.0595	−1.13439	−0.567197	+	0.823582i	$0.691972\pi$
$570$	0	0
$571$	−16.9619	−0.709834	−0.354917	−	0.934898i	$-0.615491\pi$
$571$	−16.9619	−0.709834	−0.354917	+	0.934898i	$0.615491\pi$
$572$	0	0
$573$	55.6908	2.32652
$574$	0	0
$575$	23.0967	0.963199
$576$	0	0
$577$	5.15079	0.214430	0.107215	−	0.994236i	$-0.465807\pi$
$577$	5.15079	0.214430	0.107215	+	0.994236i	$0.465807\pi$
$578$	0	0
$579$	−24.5741	−1.02126
$580$	0	0
$581$	0.0355852	0.00147632
$582$	0	0
$583$	17.0427	0.705836
$584$	0	0
$585$	−30.4157	−1.25753
$586$	0	0
$587$	−41.3209	−1.70550	−0.852748	−	0.522322i	$-0.825066\pi$
$587$	−41.3209	−1.70550	−0.852748	+	0.522322i	$0.825066\pi$
$588$	0	0
$589$	−41.9013	−1.72651
$590$	0	0
$591$	−44.4218	−1.82727
$592$	0	0
$593$	−25.3883	−1.04257	−0.521286	−	0.853382i	$-0.674547\pi$
$593$	−25.3883	−1.04257	−0.521286	+	0.853382i	$0.674547\pi$
$594$	0	0
$595$	−0.0346434	−0.00142024
$596$	0	0
$597$	−7.67500	−0.314117
$598$	0	0
$599$	−22.7053	−0.927715	−0.463857	−	0.885910i	$-0.653535\pi$
$599$	−22.7053	−0.927715	−0.463857	+	0.885910i	$0.653535\pi$
$600$	0	0
$601$	30.3212	1.23683	0.618413	−	0.785853i	$-0.287776\pi$
$601$	30.3212	1.23683	0.618413	+	0.785853i	$0.287776\pi$
$602$	0	0
$603$	15.8180	0.644160
$604$	0	0
$605$	30.6832	1.24745
$606$	0	0
$607$	25.7846	1.04656	0.523281	−	0.852160i	$-0.324708\pi$
$607$	25.7846	1.04656	0.523281	+	0.852160i	$0.324708\pi$
$608$	0	0
$609$	−0.0234754	−0.000951270 0
$610$	0	0
$611$	29.5859	1.19692
$612$	0	0
$613$	19.2973	0.779409	0.389705	−	0.920940i	$-0.372577\pi$
$613$	19.2973	0.779409	0.389705	+	0.920940i	$0.372577\pi$
$614$	0	0
$615$	4.77930	0.192720
$616$	0	0
$617$	40.5457	1.63231	0.816154	−	0.577834i	$-0.196102\pi$
$617$	40.5457	1.63231	0.816154	+	0.577834i	$0.196102\pi$
$618$	0	0
$619$	−22.3843	−0.899699	−0.449850	−	0.893104i	$-0.648522\pi$
$619$	−22.3843	−0.899699	−0.449850	+	0.893104i	$0.648522\pi$
$620$	0	0
$621$	9.94653	0.399141
$622$	0	0
$623$	0.0222508	0.000891460 0
$624$	0	0
$625$	−22.7932	−0.911727
$626$	0	0
$627$	19.6806	0.785967
$628$	0	0
$629$	−6.94628	−0.276966
$630$	0	0
$631$	−3.83787	−0.152783	−0.0763915	−	0.997078i	$-0.524340\pi$
$631$	−3.83787	−0.152783	−0.0763915	+	0.997078i	$0.524340\pi$
$632$	0	0
$633$	−3.19024	−0.126801
$634$	0	0
$635$	25.0195	0.992870
$636$	0	0
$637$	−33.7813	−1.33847
$638$	0	0
$639$	5.45364	0.215743
$640$	0	0
$641$	25.3780	1.00237	0.501185	−	0.865340i	$-0.332898\pi$
$641$	25.3780	1.00237	0.501185	+	0.865340i	$0.332898\pi$
$642$	0	0
$643$	0.995443	0.0392565	0.0196282	−	0.999807i	$-0.493752\pi$
$643$	0.995443	0.0392565	0.0196282	+	0.999807i	$0.493752\pi$
$644$	0	0
$645$	−16.0650	−0.632557
$646$	0	0
$647$	0.355590	0.0139797	0.00698985	−	0.999976i	$-0.497775\pi$
$647$	0.355590	0.0139797	0.00698985	+	0.999976i	$0.497775\pi$
$648$	0	0
$649$	−9.76750	−0.383408
$650$	0	0
$651$	−0.0566859	−0.00222170
$652$	0	0
$653$	16.1675	0.632683	0.316342	−	0.948645i	$-0.397545\pi$
$653$	16.1675	0.632683	0.316342	+	0.948645i	$0.397545\pi$
$654$	0	0
$655$	−49.1251	−1.91948
$656$	0	0
$657$	3.25110	0.126837
$658$	0	0
$659$	10.6468	0.414740	0.207370	−	0.978263i	$-0.433510\pi$
$659$	10.6468	0.414740	0.207370	+	0.978263i	$0.433510\pi$
$660$	0	0
$661$	50.6439	1.96982	0.984909	−	0.173074i	$-0.0553701\pi$
$661$	50.6439	1.96982	0.984909	+	0.173074i	$0.0553701\pi$
$662$	0	0
$663$	26.1871	1.01702
$664$	0	0
$665$	0.102961	0.00399265
$666$	0	0
$667$	−10.2276	−0.396014
$668$	0	0
$669$	−2.18608	−0.0845188
$670$	0	0
$671$	−18.3754	−0.709373
$672$	0	0
$673$	19.6510	0.757491	0.378745	−	0.925501i	$-0.376356\pi$
$673$	19.6510	0.757491	0.378745	+	0.925501i	$0.376356\pi$
$674$	0	0
$675$	12.5952	0.484789
$676$	0	0
$677$	7.65873	0.294349	0.147174	−	0.989111i	$-0.452982\pi$
$677$	7.65873	0.294349	0.147174	+	0.989111i	$0.452982\pi$
$678$	0	0
$679$	0.0188626	0.000723881 0
$680$	0	0
$681$	−38.9113	−1.49109
$682$	0	0
$683$	39.6799	1.51831	0.759155	−	0.650910i	$-0.225612\pi$
$683$	39.6799	1.51831	0.759155	+	0.650910i	$0.225612\pi$
$684$	0	0
$685$	59.0938	2.25786
$686$	0	0
$687$	13.1808	0.502879
$688$	0	0
$689$	−67.3969	−2.56762
$690$	0	0
$691$	19.6521	0.747601	0.373801	−	0.927509i	$-0.378054\pi$
$691$	19.6521	0.747601	0.373801	+	0.927509i	$0.378054\pi$
$692$	0	0
$693$	0.0105002	0.000398871 0
$694$	0	0
$695$	−13.4008	−0.508322
$696$	0	0
$697$	−1.62281	−0.0614683
$698$	0	0
$699$	−64.9249	−2.45569
$700$	0	0
$701$	25.0594	0.946480	0.473240	−	0.880934i	$-0.343084\pi$
$701$	25.0594	0.946480	0.473240	+	0.880934i	$0.343084\pi$
$702$	0	0
$703$	20.6445	0.778621
$704$	0	0
$705$	44.0200	1.65789
$706$	0	0
$707$	0.0424311	0.00159579
$708$	0	0
$709$	13.0465	0.489971	0.244985	−	0.969527i	$-0.421217\pi$
$709$	13.0465	0.489971	0.244985	+	0.969527i	$0.421217\pi$
$710$	0	0
$711$	−7.50408	−0.281425
$712$	0	0
$713$	−24.6966	−0.924894
$714$	0	0
$715$	18.9996	0.710543
$716$	0	0
$717$	−34.9206	−1.30413
$718$	0	0
$719$	24.8152	0.925450	0.462725	−	0.886502i	$-0.346872\pi$
$719$	24.8152	0.925450	0.462725	+	0.886502i	$0.346872\pi$
$720$	0	0
$721$	−0.0132736	−0.000494333 0
$722$	0	0
$723$	−11.5001	−0.427693
$724$	0	0
$725$	−12.9511	−0.480992
$726$	0	0
$727$	−17.5473	−0.650794	−0.325397	−	0.945578i	$-0.605498\pi$
$727$	−17.5473	−0.650794	−0.325397	+	0.945578i	$0.605498\pi$
$728$	0	0
$729$	−6.02541	−0.223163
$730$	0	0
$731$	5.45484	0.201755
$732$	0	0
$733$	35.1355	1.29776	0.648880	−	0.760891i	$-0.275238\pi$
$733$	35.1355	1.29776	0.648880	+	0.760891i	$0.275238\pi$
$734$	0	0
$735$	−50.2622	−1.85395
$736$	0	0
$737$	−9.88095	−0.363969
$738$	0	0
$739$	48.7312	1.79261	0.896303	−	0.443442i	$-0.146243\pi$
$739$	48.7312	1.79261	0.896303	+	0.443442i	$0.146243\pi$
$740$	0	0
$741$	−77.8286	−2.85911
$742$	0	0
$743$	30.0116	1.10102	0.550510	−	0.834829i	$-0.314433\pi$
$743$	30.0116	1.10102	0.550510	+	0.834829i	$0.314433\pi$
$744$	0	0
$745$	−43.7302	−1.60215
$746$	0	0
$747$	−15.7839	−0.577504
$748$	0	0
$749$	0.0812713	0.00296959
$750$	0	0
$751$	6.30898	0.230218	0.115109	−	0.993353i	$-0.463278\pi$
$751$	6.30898	0.230218	0.115109	+	0.993353i	$0.463278\pi$
$752$	0	0
$753$	45.6542	1.66373
$754$	0	0
$755$	−32.4229	−1.17999
$756$	0	0
$757$	−18.7315	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$757$	−18.7315	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$758$	0	0
$759$	11.5997	0.421043
$760$	0	0
$761$	5.56942	0.201891	0.100946	−	0.994892i	$-0.467813\pi$
$761$	5.56942	0.201891	0.100946	+	0.994892i	$0.467813\pi$
$762$	0	0
$763$	0.0555398	0.00201067
$764$	0	0
$765$	15.3662	0.555565
$766$	0	0
$767$	38.6265	1.39472
$768$	0	0
$769$	−25.3423	−0.913868	−0.456934	−	0.889501i	$-0.651052\pi$
$769$	−25.3423	−0.913868	−0.456934	+	0.889501i	$0.651052\pi$
$770$	0	0
$771$	62.2129	2.24054
$772$	0	0
$773$	−48.4648	−1.74316	−0.871579	−	0.490255i	$-0.836904\pi$
$773$	−48.4648	−1.74316	−0.871579	+	0.490255i	$0.836904\pi$
$774$	0	0
$775$	−31.2730	−1.12336
$776$	0	0
$777$	0.0279288	0.00100194
$778$	0	0
$779$	4.82302	0.172803
$780$	0	0
$781$	−3.40669	−0.121901
$782$	0	0
$783$	−5.57736	−0.199319
$784$	0	0
$785$	52.2825	1.86604
$786$	0	0
$787$	−2.77234	−0.0988234	−0.0494117	−	0.998778i	$-0.515735\pi$
$787$	−2.77234	−0.0988234	−0.0494117	+	0.998778i	$0.515735\pi$
$788$	0	0
$789$	−25.0052	−0.890208
$790$	0	0
$791$	0.0138112	0.000491070 0
$792$	0	0
$793$	72.6670	2.58048
$794$	0	0
$795$	−100.278	−3.55648
$796$	0	0
$797$	15.2952	0.541782	0.270891	−	0.962610i	$-0.412682\pi$
$797$	15.2952	0.541782	0.270891	+	0.962610i	$0.412682\pi$
$798$	0	0
$799$	−14.9470	−0.528786
$800$	0	0
$801$	−9.86940	−0.348718
$802$	0	0
$803$	−2.03084	−0.0716668
$804$	0	0
$805$	0.0606849	0.00213886
$806$	0	0
$807$	35.3684	1.24503
$808$	0	0
$809$	−24.1501	−0.849071	−0.424536	−	0.905411i	$-0.639563\pi$
$809$	−24.1501	−0.849071	−0.424536	+	0.905411i	$0.639563\pi$
$810$	0	0
$811$	−31.7152	−1.11367	−0.556835	−	0.830623i	$-0.687984\pi$
$811$	−31.7152	−1.11367	−0.556835	+	0.830623i	$0.687984\pi$
$812$	0	0
$813$	−42.9239	−1.50541
$814$	0	0
$815$	−75.3438	−2.63918
$816$	0	0
$817$	−16.2119	−0.567183
$818$	0	0
$819$	−0.0415241	−0.00145097
$820$	0	0
$821$	−8.02947	−0.280230	−0.140115	−	0.990135i	$-0.544747\pi$
$821$	−8.02947	−0.280230	−0.140115	+	0.990135i	$0.544747\pi$
$822$	0	0
$823$	−8.70556	−0.303457	−0.151728	−	0.988422i	$-0.548484\pi$
$823$	−8.70556	−0.303457	−0.151728	+	0.988422i	$0.548484\pi$
$824$	0	0
$825$	14.6886	0.511391
$826$	0	0
$827$	18.0910	0.629085	0.314542	−	0.949243i	$-0.398149\pi$
$827$	18.0910	0.629085	0.314542	+	0.949243i	$0.398149\pi$
$828$	0	0
$829$	−45.2705	−1.57231	−0.786154	−	0.618031i	$-0.787931\pi$
$829$	−45.2705	−1.57231	−0.786154	+	0.618031i	$0.787931\pi$
$830$	0	0
$831$	−41.8689	−1.45242
$832$	0	0
$833$	17.0665	0.591319
$834$	0	0
$835$	1.65741	0.0573571
$836$	0	0
$837$	−13.4676	−0.465510
$838$	0	0
$839$	13.3138	0.459645	0.229823	−	0.973233i	$-0.426185\pi$
$839$	13.3138	0.459645	0.229823	+	0.973233i	$0.426185\pi$
$840$	0	0
$841$	−23.2650	−0.802243
$842$	0	0
$843$	−15.3762	−0.529584
$844$	0	0
$845$	−33.1955	−1.14196
$846$	0	0
$847$	0.0418894	0.00143934
$848$	0	0
$849$	36.0077	1.23578
$850$	0	0
$851$	12.1678	0.417108
$852$	0	0
$853$	1.89390	0.0648461	0.0324230	−	0.999474i	$-0.489678\pi$
$853$	1.89390	0.0648461	0.0324230	+	0.999474i	$0.489678\pi$
$854$	0	0
$855$	−45.6685	−1.56183
$856$	0	0
$857$	45.8339	1.56566	0.782828	−	0.622238i	$-0.213776\pi$
$857$	45.8339	1.56566	0.782828	+	0.622238i	$0.213776\pi$
$858$	0	0
$859$	−7.96693	−0.271828	−0.135914	−	0.990721i	$-0.543397\pi$
$859$	−7.96693	−0.271828	−0.135914	+	0.990721i	$0.543397\pi$
$860$	0	0
$861$	0.00652480	0.000222364 0
$862$	0	0
$863$	42.7741	1.45605	0.728023	−	0.685552i	$-0.240439\pi$
$863$	42.7741	1.45605	0.728023	+	0.685552i	$0.240439\pi$
$864$	0	0
$865$	−0.179347	−0.00609800
$866$	0	0
$867$	24.6064	0.835679
$868$	0	0
$869$	4.68752	0.159013
$870$	0	0
$871$	39.0751	1.32401
$872$	0	0
$873$	−8.36657	−0.283165
$874$	0	0
$875$	0.00579828	0.000196018 0
$876$	0	0
$877$	17.4478	0.589169	0.294585	−	0.955625i	$-0.404819\pi$
$877$	17.4478	0.589169	0.294585	+	0.955625i	$0.404819\pi$
$878$	0	0
$879$	35.4661	1.19624
$880$	0	0
$881$	−1.78522	−0.0601455	−0.0300728	−	0.999548i	$-0.509574\pi$
$881$	−1.78522	−0.0601455	−0.0300728	+	0.999548i	$0.509574\pi$
$882$	0	0
$883$	−52.8617	−1.77894	−0.889469	−	0.456995i	$-0.848926\pi$
$883$	−52.8617	−1.77894	−0.889469	+	0.456995i	$0.848926\pi$
$884$	0	0
$885$	57.4711	1.93187
$886$	0	0
$887$	44.9399	1.50893	0.754467	−	0.656338i	$-0.227895\pi$
$887$	44.9399	1.50893	0.754467	+	0.656338i	$0.227895\pi$
$888$	0	0
$889$	0.0341572	0.00114559
$890$	0	0
$891$	13.4777	0.451519
$892$	0	0
$893$	44.4227	1.48655
$894$	0	0
$895$	−80.6332	−2.69527
$896$	0	0
$897$	−45.8721	−1.53162
$898$	0	0
$899$	13.8482	0.461864
$900$	0	0
$901$	34.0492	1.13434
$902$	0	0
$903$	−0.0219322	−0.000729857 0
$904$	0	0
$905$	−8.42273	−0.279981
$906$	0	0
$907$	−56.3253	−1.87025	−0.935126	−	0.354316i	$-0.884714\pi$
$907$	−56.3253	−1.87025	−0.935126	+	0.354316i	$0.884714\pi$
$908$	0	0
$909$	−18.8204	−0.624234
$910$	0	0
$911$	39.0206	1.29281	0.646406	−	0.762994i	$-0.276271\pi$
$911$	39.0206	1.29281	0.646406	+	0.762994i	$0.276271\pi$
$912$	0	0
$913$	9.85964	0.326306
$914$	0	0
$915$	108.119	3.57430
$916$	0	0
$917$	−0.0670666	−0.00221473
$918$	0	0
$919$	21.4208	0.706608	0.353304	−	0.935509i	$-0.385058\pi$
$919$	21.4208	0.706608	0.353304	+	0.935509i	$0.385058\pi$
$920$	0	0
$921$	11.0678	0.364695
$922$	0	0
$923$	13.4721	0.443438
$924$	0	0
$925$	15.4080	0.506612
$926$	0	0
$927$	5.88752	0.193372
$928$	0	0
$929$	−41.2902	−1.35469	−0.677345	−	0.735666i	$-0.736869\pi$
$929$	−41.2902	−1.35469	−0.677345	+	0.735666i	$0.736869\pi$
$930$	0	0
$931$	−50.7220	−1.66235
$932$	0	0
$933$	27.8046	0.910283
$934$	0	0
$935$	−9.59867	−0.313910
$936$	0	0
$937$	−39.7702	−1.29924	−0.649618	−	0.760261i	$-0.725071\pi$
$937$	−39.7702	−1.29924	−0.649618	+	0.760261i	$0.725071\pi$
$938$	0	0
$939$	39.3402	1.28382
$940$	0	0
$941$	−14.6278	−0.476852	−0.238426	−	0.971161i	$-0.576631\pi$
$941$	−14.6278	−0.476852	−0.238426	+	0.971161i	$0.576631\pi$
$942$	0	0
$943$	2.84268	0.0925704
$944$	0	0
$945$	0.0330930	0.00107651
$946$	0	0
$947$	17.7843	0.577911	0.288956	−	0.957343i	$-0.406692\pi$
$947$	17.7843	0.577911	0.288956	+	0.957343i	$0.406692\pi$
$948$	0	0
$949$	8.03114	0.260702
$950$	0	0
$951$	−59.7522	−1.93760
$952$	0	0
$953$	−7.43982	−0.240999	−0.120500	−	0.992713i	$-0.538450\pi$
$953$	−7.43982	−0.240999	−0.120500	+	0.992713i	$0.538450\pi$
$954$	0	0
$955$	80.7251	2.61220
$956$	0	0
$957$	−6.50435	−0.210256
$958$	0	0
$959$	0.0806760	0.00260517
$960$	0	0
$961$	2.43923	0.0786848
$962$	0	0
$963$	−36.0481	−1.16163
$964$	0	0
$965$	−35.6207	−1.14667
$966$	0	0
$967$	30.5279	0.981712	0.490856	−	0.871241i	$-0.336684\pi$
$967$	30.5279	0.981712	0.490856	+	0.871241i	$0.336684\pi$
$968$	0	0
$969$	39.3194	1.26312
$970$	0	0
$971$	2.73108	0.0876445	0.0438222	−	0.999039i	$-0.486046\pi$
$971$	2.73108	0.0876445	0.0438222	+	0.999039i	$0.486046\pi$
$972$	0	0
$973$	−0.0182950	−0.000586512 0
$974$	0	0
$975$	−58.0873	−1.86028
$976$	0	0
$977$	35.7060	1.14233	0.571167	−	0.820834i	$-0.306491\pi$
$977$	35.7060	1.14233	0.571167	+	0.820834i	$0.306491\pi$
$978$	0	0
$979$	6.16506	0.197036
$980$	0	0
$981$	−24.6348	−0.786529
$982$	0	0
$983$	54.4981	1.73822	0.869110	−	0.494619i	$-0.164692\pi$
$983$	54.4981	1.73822	0.869110	+	0.494619i	$0.164692\pi$
$984$	0	0
$985$	−64.3904	−2.05165
$986$	0	0
$987$	0.0600970	0.00191291
$988$	0	0
$989$	−9.55527	−0.303840
$990$	0	0
$991$	−20.9596	−0.665805	−0.332903	−	0.942961i	$-0.608028\pi$
$991$	−20.9596	−0.665805	−0.332903	+	0.942961i	$0.608028\pi$
$992$	0	0
$993$	16.6019	0.526847
$994$	0	0
$995$	−11.1251	−0.352689
$996$	0	0
$997$	22.7228	0.719640	0.359820	−	0.933022i	$-0.382838\pi$
$997$	22.7228	0.719640	0.359820	+	0.933022i	$0.382838\pi$
$998$	0	0
$999$	6.63541	0.209935

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.w.1.4		29
4.3	odd	2		4024.2.a.e.1.26	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.26	✓	29	4.3	odd	2
8048.2.a.w.1.4		29	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-2.22567 q^{3} -3.22615 q^{5} -0.00440441 q^{7} +1.95359 q^{9} +O(q^{10})\)	\(q-2.22567 q^{3} -3.22615 q^{5} -0.00440441 q^{7} +1.95359 q^{9} -1.22033 q^{11} +4.82592 q^{13} +7.18034 q^{15} -2.43808 q^{17} +7.24602 q^{19} +0.00980273 q^{21} +4.27079 q^{23} +5.40806 q^{25} +2.32897 q^{27} -2.39478 q^{29} -5.78267 q^{31} +2.71605 q^{33} +0.0142093 q^{35} +2.84908 q^{37} -10.7409 q^{39} +0.665610 q^{41} -2.23735 q^{43} -6.30257 q^{45} +6.13063 q^{47} -6.99998 q^{49} +5.42635 q^{51} -13.9656 q^{53} +3.93698 q^{55} -16.1272 q^{57} +8.00396 q^{59} +15.0576 q^{61} -0.00860439 q^{63} -15.5692 q^{65} +8.09692 q^{67} -9.50535 q^{69} +2.79160 q^{71} +1.66417 q^{73} -12.0365 q^{75} +0.00537485 q^{77} -3.84118 q^{79} -11.0443 q^{81} -8.07946 q^{83} +7.86561 q^{85} +5.32998 q^{87} -5.05194 q^{89} -0.0212553 q^{91} +12.8703 q^{93} -23.3768 q^{95} -4.28267 q^{97} -2.38403 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more